Kemperman's inequality and Freiman's lemma via few translates

Abstract

Let $G$ be a connected compact group equipped with the normalised Haar measure $\mu$. Our first result shows that given $\alpha ,\beta >0$, there is a constant $c=c(\alpha ,\beta )>0$ such that for any compact sets $A,B\subseteq G$ with $\alpha \mu (B)\ge \mu (A)\ge \mu (B)$ and $\mu (A)+\mu (B)\le 1-\beta$, there exist $b_1,\dots b_c\in B$ such that \[ μ(A\cdot \{b_1,\dots,b_c\})\geq μ(A)+μ(B).\] A special case of this, that is, when $G={\mathbb{T}}^d$, confirms a recent conjecture of Bollobás, Leader and Tiba. We also prove a quantitatively stronger version of such a result in the discrete setting of ${\mathbb{R}}^d$. Thus, given $d\in \mathbb{N}$, we show that there exists $c=c(d)>0$ such that for any finite, non-empty set $A\subseteq {\mathbb{R}}^d$ which is not contained in a translate of a hyperplane, one can find $a_1,\dots ,a_c\in A$ satisfying \[ |A+ \{a_1, \dots, a_c\}| \geq (d+1)|A| - O_d(1). \] The main term here is optimal and recovers the bounds given by Freiman's lemma up to the $O_d(1)$ error term.

arXiv comment18 pages; typos corrected, the error term in Theorem 1.2 improved